Entropy-based Exploration Conduction for Multi-step Reasoning

• LLMs may over- or under-reason when their exploration depth does not match expectations for a problem.The paper attributes this mismatch partly to the absence of evaluation and regulatory mechanisms for the reasoning process.Source: §1 · p. 1
• Depth mismatch can produce inaccurate, insufficient, or redundant reasoning outcomes and unnecessary computation costs.Source: §1 · p. 1
• Reasoning needs vary across tasks, and models may not accurately judge and adjust exploration depth from pretraining knowledge alone.Source: §1 · p. 1

• Outcome-based optimization uses post-training, while process-based methods use self-critique or step labels and can be difficult to interpret.The paper describes outcome-based approaches as resource-intensive and process-based approaches as immediate and low-cost, but potentially opaque.Source: §3 · p. 2

Our Target

• The method aims to determine exploration depth from task information and reasoning state while reducing unnecessary exploration.Source: §3 · p. 2

Our Method

• Entro-duction uses entropy changes to assess reasoning uncertainty and update the probability distribution for the next exploratory action.The available actions include deepening, stopping, or expanding exploration.Source: §3 · p. 2
• Its behavior-selection mechanism refers to the LLM’s expectancy and confidence during reasoning.Source: §3 · p. 2

• The paper reports experiments across various reasoning tasks and states that Entro-duction outperforms baseline methods.Source: §3 · p. 2

Reasoning Steps and Structures

• CoT-SC, Complex CoT, and ToT extend CoT-style reasoning structures to explore solution spaces in multiple directions.Source: §2 · p. 2
• The breadth and depth of reasoning structures depend heavily on predefined settings and vary across tasks, limiting generalizability and flexibility.Source: §2 · p. 3

Optimization of Reasoning Depth

• Automated depth optimization includes reinforcement-learning approaches and methods that dynamically adjust exploration during reasoning.Source: §2 · p. 3
• These methods may lack precision, harm reasoning when task characteristics are not understood, or rely on unsuitable preset structures for novel tasks.The paper links these issues to reduced reliability and efficiency, inaccurate outcomes, or redundant exploration.Source: §2 · p. 3

3.1 Problem Formulation

• The reasoning structure contains directed links between sentence-level nodes, and its conclusion is determined by voting across valid chains.The target is a structure with optimal reasoning accuracy and depth.Source: §3.1 · p. 3

3.2 Reasoning State Evaluation

• Entropy measures uncertainty from token probabilities, while normalized entropy supports comparisons across steps of different lengths and variance entropy tracks uncertainty fluctuations across steps.The normalized entropy is bounded between 0 and 1; variance entropy indicates consistency or divergence in the thought process.Source: §3.2 · pp. 3–4

3.3 Exploration Behaviors

• Changes in entropy and variance entropy indicate whether reasoning is becoming more certain, broader, or less stable, guiding the probability of exploration behaviors.The described cases include continuing in the current direction or considering exploration in different directions when step-to-step fluctuations increase.Source: §3.3 · pp. 4–5
• The available behaviors are deepen, which extends a chain; expand, which branches it into two chains; and stop, which terminates its extension.These behaviors respectively add a node, create two chains from a split point, or end the chain at its current node.Source: §3.3 · p. 5

3.4 Behavior Selection Mechanism

• At each reasoning step, the entropy-change pair forms a state that maps to a preferred action, which is sampled according to an epsilon-greedy rule.If the sampled action is Stop, reasoning ends; otherwise, the system transitions to an updated state.Source: §3.4 · p. 5, Algorithm 1 Entro-duction · p. 5

Algorithm 1 Entro-duction

• Algorithm 1 initializes reasoning chains, evaluates their entropy measures, samples and executes actions for active chains, and returns a final conclusion via consensus.Its stated inputs include the task, reasoner, maximum steps, and exploration rate.Source: Algorithm 1 Entro-duction · p. 5

4.1 Experiment Settings

• Evaluation uses Llama-3.1-8B-Instruct on GSM8K, SVAMP, StrategyQA, and CSQA, comparing reasoning structures and depth-optimization methods by accuracy and reasoning-step counts.Step counts are additionally measured for reasoning structures.Source: §4.1 · p. 6, §4.2 · p. 6

4.2 Overall Performance

• Across four datasets, Entro-duction beats most baselines; GSM8K accuracy is 0.85 versus CoT 0.75, CoT-SC@maj64 0.80, and Complex CoT 0.81.It uses far fewer steps than tree methods and achieves higher accuracy with similar or slightly increased steps than CoT and Complex CoT; it also surpasses Self-talk and often DRR.Source: §4.2 · pp. 6–7, Table 1 · p. 7

4.3.1 Impact of Jointly Using Entropy and Variance Entropy

• Entropy alone tends to stop prematurely in Scenario 2, while variance entropy alone slightly outperforms it but remains inadequate in Scenario 3.The study compares settings using neither signal, either signal alone, or both.Source: §4.3.1 · p. 7, Figure 3 · p. 7

4.3.2 Impact of Expansion in Reasoning

• Enabling Expand generally improves accuracy over using only Deepen and Stop, particularly on SVAMP and StrategyQA, where multiple reasoning paths matter.The authors interpret expansion as covering more potential directions and improving reasoning completeness.Source: §4.3.2 · p. 7, Figure 4 · p. 8

4.3.3 Impact of Soft Stop

• Hard stopping performs worst and soft stopping best across four datasets; Stop@2 matches or beats Stop@3 on SVAMP and CSQA, supporting two to three extra steps.Hard stopping may end exploration before the correct conclusion; the soft-stop settings add one or two reasoning steps after the stop signal.Source: §4.3.3 · pp. 7–8, Figure 5 · p. 8

4.4 Robustness Study

• ϵ = 0.25 yields the highest accuracy on all four datasets; ϵ = 0.5 beats ϵ = 0.1 on math but underperforms on commonsense, while ϵ = 0.05 performs poorly.The authors associate insufficient exploration with the low-ϵ result and warn that over-exploration may distract commonsense reasoning from the question background.Source: Figure 6 · p. 8, §4.4 · p. 8

5 Conclusion

• Entro-duction dynamically adjusts exploration depth during multi-step reasoning by monitoring entropy and variance entropy.The conclusion says experiments across multiple reasoning datasets demonstrate the method’s effectiveness and that of its components.Source: §5 · p. 8

Limitations

• Without precise calibration, the method might produce suboptimal reasoning and may not balance exploration and exploitation adaptively in real time.Source: Limitations · p. 9
• Evaluation on four benchmark datasets and one Llama model may not provide a comprehensive view of generalization across models with different sizes and pretraining processes.Source: Limitations · p. 9
• The evaluated tasks may not fully reflect real-world scenarios with variable reasoning tasks and more complex, external solution spaces.Source: Limitations · p. 9

A.1 Baselines

• CoT uses the prompt “Let’s think step-by-step,” while CoT-SC uses majority voting with the same few-shot examples as standard CoT.Source: §A.1 · p. 12
• Complex CoT follows Zhou et al. (2022), while Self-talk and DRR follow Yang et al. (2024).Source: §A.1 · p. 12

A.2 Answer-Cleaning

• For GSM8K, answer cleansing removes commas, extracts numbers with regular expressions, then selects the first or last number according to predefined dataset logic.The selected value becomes the cleansed prediction used for evaluation against the ground truth.Source: §A.2 · p. 12